Vizing's celebrated theorem states that every simple graph with maximum degree $\Delta$ admits a $(\Delta+1)$ edge coloring which can be found in $O(m \cdot n)$ time on $n$-vertex $m$-edge graphs. This is just one color more than the trivial lower bound of $\Delta$ colors needed in any proper edge coloring. After a series of simplifications and variations, this running time was eventually improved by Gabow, Nishizeki, Kariv, Leven, and Terada in 1985 to $O(m\sqrt{n\log{n}})$ time. This has effectively remained the state-of-the-art modulo an $O(\sqrt{\log{n}})$-factor improvement by Sinnamon in 2019. As our main result, we present a novel randomized algorithm that computes a $\Delta+O(\log{n})$ coloring of any given simple graph in $O(m\log{\Delta})$ expected time; in other words, a near-linear time randomized algorithm for a ``near''-Vizing's coloring. As a corollary of this algorithm, we also obtain the following results: * A randomized algorithm for $(\Delta+1)$ edge coloring in $O(n^2\log{n})$ expected time. This is near-linear in the input size for dense graphs and presents the first polynomial time improvement over the longstanding bounds of Gabow et.al. for Vizing's theorem in almost four decades. * A randomized algorithm for $(1+\varepsilon) \Delta$ edge coloring in $O(m\log{(1/\varepsilon)})$ expected time for any $\varepsilon = \omega(\log{n}/\Delta)$. The dependence on $\varepsilon$ exponentially improves upon a series of recent results that obtain algorithms with runtime of $\Omega(m/\varepsilon)$ for this problem.

翻译：Vizing著名定理指出，任何最大度为Δ的简单图均存在(Δ+1)边着色方案，且该方案可在具有n个顶点、m条边的图上以O(m·n)时间求解。这仅比任何正常边着色所需颜色数的平凡下界Δ多一种颜色。经过一系列简化与变体改进，Gabow、Nishizeki、Kariv、Leven和Terada于1985年最终将运行时间提升至O(m√(n log n))。除Sinnamon在2019年取得的O(√(log n))因子改进外，该结果实质上一直保持最优。作为主要研究成果，我们提出一种新颖的随机算法，可在O(m log Δ)期望时间内为任意给定简单图计算Δ+O(log n)着色方案；换言之，这是一种实现"近"Vizing着色的近线性时间随机算法。基于该算法，我们同时获得以下结果：* 可在O(n² log n)期望时间内实现(Δ+1)边着色的随机算法。对于稠密图而言该时间复杂度接近输入规模的线性阶，这标志着近四十年来对Gabow等人提出的Vizing定理时间复杂度界限的首个多项式级别改进。* 对于任意满足ε=ω(log n/Δ)的ε，可在O(m log(1/ε))期望时间内实现(1+ε)Δ边着色的随机算法。该算法对ε的依赖关系较近期一系列针对该问题提出的Ω(m/ε)时间复杂度算法实现了指数级改进。